Friedrich's cylinder at spatial infinity. In [1]}
Friedrich puts forward an alternative conformal representation of the region of
spacetime in the neighbourhood of spatial infinity —the F-gauge. The aim of this
representation is the formulation of a regular initial value
problem at spatial infinity for the conformal Einstein field
equations. This initial value problem is key in the programme
to analyse the genericity of asymptotically simple spacetimes and
relies heavily on the properties of certain conformal invariants
(conformal geodesics) and it is such that both the equations
and the initial data are regular at the conformal boundary. A central
structure in this representation of spatial infinity is the
cylinder at spatial infinity which, in broad terms, corresponds
to the blow-up of the traditional point at spatial infinity to a
2-sphere plus a time dimension —hence, the cylinder at spatial
infinity. In contrast to the hyperboloid at spatial infinity, the cylinder at spatial infinity has a finite extension in the time
direction. The endpoints of the cylinder, the critical sets
correspond to the points where spatial infinity meets null
infinity. Thus, the F-gauge is ideally suited to the analysis of the connection between the behaviour of the gravitational field at spatial infinity and radiative properties. This idea has been elaborated
in [2]} to express the Newman-Penrose constants in terms of
the initial data for the Einstein field equation and to study the behaviour of the Bondi mass as one approaches spatial infinity —see
[3]}; also [4]}. The key property of the cylinder at
spatial infinity which allows connecting properties of the Cauchy initial
data in a neighbourhood of spatial infinity with the behaviour near
null infinity is that the cylinder at spatial infinity is a total
characteristic of the conformal Einstein field equations —that is,
the associated evolution equations reduce in its entirety to a system
of transport equations on this hypersurface. This property precludes
the possibility of prescribing boundary data on the cylinder but
allows computing a particular type of asymptotic expansions which
make it possible to understand the role of certain pieces of the
initial data have on the regularity properties of the gravitational at
null infinity —the so-called peeling behaviour; see
[1]}, [2]}, [7]}, [8]}, [9]}, [10]}, [11]}.
